Midrash derives from the verb derash ‘to seek, ask’. Already in Scripture, the verb is used with primarily theological connotations, with God or the Torah, etc. as object (e.g. Ezra 7.10 ‘studying God’s law’; Isa 34.16 ‘searching in God’s book’). The noun ‘midrash’ appears in two late passages: according to 2 Chr 13.22, the history of Abijah is recorded ‘in the midrash of the prophet Iddo’; 24.27 speaks of the ‘midrash on the Book of Kings’. The precise meaning of midrash in both passages is not certain: whether ‘book, work’ (LXX translates biblion or graphé, the Vulgate liber) or already in the later sense of an ‘interpretative writing’. Sir 51.23 is the first instance of beth midrash, ‘house of teaching, schoolhouse’. In the sense of ‘teaching, instruction’, the term is now documented at Qumran, too: midrash le-maskil occurs in 4QSb and 4QSd; the Scrolls also frequently use darash as ‘to search out, interpret’ (the laws or commandments: 1QS 5.11; 6.6; 4QFlor 1.11) and speak of the midrash ha-torah (1QS 8.15; CD 20.6).
This is already equivalent to rabbinic usage, where midrash means especially ‘research, study’ and is distinguished, as ‘theory’, from the more essential ‘practice’ (ma’aseh): Abot 1.17. In this regard, it is synonymous with talmud, which is contrasted with practice, e.g. in p.Pes 3.7, 30b. In the narrower sense of ‘interpretation’, Ket 4.6 says zeh midrash darash, ‘he presented this interpretation’ (the objet of interpretation here is the Ketubbah). Midrash is more particularly applied to the occupation with the Bible, thus e.g. p.Yoma 3.5, 40c, according to which every interpretation of Scripture (midrash) must address the content. The beth ha-midrash, therefore, is the house of study, especially of the Bible (e.g. Shab 16.1; Pes 4.4). Midrash also comes to designate more specifically the result of interpretation or writings containing biblical interpretation. The darshan (Aram. darosha) is the Bible interpreter or preacher.
H.L. Strack and Günter Stemberger, Introduction to the Talmud and Midrash, trans. and ed. Markus Bockmuehl, 2nd ed. (Minneapolis: Fortress Press, 1996), 234-235.